How can I delete or shift certain sections of the Timeline?
In the Dope Sheet editor, you can select a specific keyframe, or several of them (use B to border select), and move them with G. You can delete the keys with Suppr or X.

To go further, maybe you could create an Action for each slide, that's a great way to manipulate them using the NLA editor (basically, each slide will act like a movie clip in a video editor). NLA Editor Action Editor